I know that in general norms in a topological vector space are continous maps, that is to say:
$$\lim_{n\to \infty} \|x_n\|=\|\lim_{n\to\infty} x_n \|$$
But this is not always the case in $L^1(\mathbb{R})$ (because we cannot always swap integral and limit). I know that I'm missing something stupid but please help me.